A-POLYNOMIAL AND BLOCH INVARIANTS OF HYPERBOLIC 3-MANIFOLDS

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1 A-POLYNOMIAL AND BLOCH INVARIANTS OF HYPERBOLIC 3-MANIFOLDS ABHIJIT CHAMPANERKAR Abstract. Let N be a complete, orientable, finite-volume, one-cusped hyperbolic 3-manifold with an ideal triangulation. Using combinatorics of the ideal triangulation of N we construct a plane curve in C C which contains the squares of eigenvalues of PSL(2, C) representations of the meridian and longitude. We show that the defining polynomial of this curve is related to the PSL(2, C) A-polynomial and has properties similar to the classical A-polynomial. We further show that a factor of this polynomial, A 0(l, m), associated to the discrete, faithful representation of π 1(N) in PSL(2, C) is independent of the ideal triangulation. The Bloch invariant β(n) of N is related to the volume and Chern-Simons invariant of N. The variation of Bloch invariant is defined to be the change of β(n) under Dehn surgery on N. We relate A 0(l, m) to the variation of the Bloch invariant of N. We show that A 0(l, m) determines the variation of Bloch invariant in the case when A 0(l, m) is a defining equation of a rational curve. We also show that in this case the Bloch invariant reads the symmetry of A 0(l, m). 1. Introduction This paper is divided into 2 parts. In the first part we study a two variable polynomial arising from the combinatorics of ideal triangulations of a cusped hyperbolic 3-manifold. Next we study the variation of Bloch invariants of hyperbolic 3-manifolds and relate it to this polynomial. Essential surfaces are central to understanding the topology of 3-manifolds. Culler and Shalen [14] studied character varieties of SL(2, C) representations of 3-manifolds groups and used them to detect essential surfaces in 3-manifolds. The authors of [7] constructed a plane curve using representations of the meridian and longitude in SL(2, C) and using ideas developed in [14] showed that this plane curve carried topological information about essential surfaces in a 3-manifold with torus boundary. The A-polynomial is the defining equation of this plane curve. The A-polynomial provided a computable tool to study information obtained from SL(2, C) character varieties. It has many interesting properties and has emerged as an important tool in studying the topology of 3-manifolds. In an analogous manner the PSL(2, C) character 2000 Mathematics Subject Classification. Primary 57M50; Secondary 57N10. 1

2 2 Abhijit Champanerkar variety was studied in [3]. For more details on SL(2, C) character varieties see [13], [14] and [35] and for A-polynomials see [7], [8], [9], [10]. Ideal triangulations are combinatorial triangulations with vertices at infinity. Ideal triangulations arise naturally in the study of of cusped hyperbolic 3- manifolds and play an important role in their study. Thurston [38] showed that any hyperbolic 3-manifold is obtainable from a cusped one by Dehn surgeries on some of the cusps. Epstein and Penner [20] showed that any cusped hyperbolic 3-manifold can be decomposed into ideal polyhedra. Ideal triangulations arise in practice, e.g. in the program SnapPea [40] written by Jeff Weeks for studying hyperbolic 3-manifolds. Ideal triangulations have been used to study geometric invariants like volume [31], Chern-Simons [26], arithmetic invariants like the invariant trace field [28], [34] and Bloch invariants [30], essential surfaces in hyperbolic 3-manifolds [39], [43] and Dehn surgeries on cusps of hyperbolic 3-manifolds [15]. We construct a two variable polynomial using ideal triangulations as follows. Let N be a one-cusped, complete, orientable, finite-volume hyperbolic 3-manifold having an ideal triangulation T with n tetrahedra. N is homeomorphic to the interior of a 3-manifold with torus boundary. Using the combinatorics of the triangulation developed in [31] one obtains n gluing equations and 2 completeness equations in the cross-ratio parameters z 1,...,z n of the ideal tetrahedra, considered as hyperbolic ideal tetrahedra. The complete hyperbolic structure on N is given by parameters z 0 1,...,z0 n which satisfy the gluing and completeness equations and lie in the upper-half plane. Let P(N) = {(z 1,...,z n,t) C n+1 : (z 1,...,z n ) satisfy gluing equations and n t z i (1 z i ) = 1} We call P(N) the Parameter Space of N depending on the triangulation T. The coordinate t ensures that the degenerate values of the parameters (i.e.z i = 0,1) do not occur in P(N). The completeness equations give the square of eigenvalues of the meridian and longitude in the holonomy representation of π 1 (N) to PSL(2, C) as rational functions in z i s. Using this we define a map Hol: P(N) C C given by Hol(z,t) = (l(z),m(z)). The image of Hol is a curve and we call it the Holonomy Variety of N and denote it by H(N). The defining polynomial of H(N) is denoted by H(l,m). We define the PSL(2, C) analog of the classical A-polynomial as the defining polynomial of the curve containing the squares of eigenvalues of representations of the meridian and longitude in PSL(2, C) which extend to representations of π 1 (N). Let us denote this polynomial by A(l,m). We show:

3 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 3 Theorem H(l, m) divides the PSL(2, C) A-polynomial A(l, m). Let P 0 (N) denote the component of the Parameter Space containing the parameter z = (z 0 1,... z0 n) for the complete structure. Let H 0 (N) be the image of P 0 (N) under the map Hol and let H 0 (l,m) denote the corresponding factor of H(l, m). Since the complete hyperbolic structure can also be described as a discrete, faithful representation in PSL(2, C), we obtain a similar factor of A(l,m) which we denote by A 0 (l,m). We show: Theorem H 0 (l,m) = A 0 (l,m) and hence the polynomial H 0 (l,m) is independent of the ideal triangulation of N. We will denote H 0 (l,m) by A 0 (l,m) from now on. The Newton polygon of a polynomial p(x,y) = c mn x m y n is the convex hull of the points (m,n) Z Z where c mn 0. If S is an incompressible surface with non-empty boundary in N then S is a family of simple closed curves on N and hence determine a homology class in H 1 ( N; Z) given by pm+ql. The boundary slope of S is the rational number p/q. It is proved in [7] that the slopes of the sides of the Newton polygon of the A-polynomial are boundary slopes of essential surfaces in the 3-manifold. Theorem The slopes of the sides of the Newton Polygon of H(l,m) are boundary slopes of incompressible surfaces in M which correspond to ideal points of H(N). The terms of a two variable polynomial appearing along an edge of its Newton polygon may be viewed as a polynomial in a single variable called an edge polynomial. Another interesting property of the A-polynomial proved in [7] is that its edge polynomials are cyclotomic. Using a K- theoretic argument similar to the one given in [7] we show Theorem H(l, m) has cyclotomic edge polynomials. It follows from the work of Thurston [38] that hyperbolic geometry is prevalent in 3-manifolds and understanding geometric invariants is an important tool in the study of 3-manifolds. Bloch invariants for hyperbolic 3-manifolds were introduced by Neumann and Yang in [30]. Bloch invariants capture geometric information such as volume, Chern-Simons invariant and scissors congruence information of the hyperbolic 3-manifold. C {0, 1} is the cross

4 4 Abhijit Champanerkar ratio parameter space for non-degenerate, ordered hyperbolic, ideal tetrahedra up to isometry. The pre-bloch group P(C) is defined as P(C) = Z(C {0, 1})/(5-term relations) where 5-term relations relate the parameters of tetrahedra when a hyperbolic polytope on five ideal vertices is decomposed as two ideal tetrahedra with a common face or as three ideal tetrahedra with a common edge. P(C) is the orientation sensitive analog of the scissors congruence group of H 3. The analog of the Dehn invariant for scissors congruence is a map µ : P(C) C C defined by µ([z]) = 2 (z (1 z)). The Bloch group of C is defined to be ker(µ) and denoted as B(C). A well known conjecture is Conjecture (Bloch Rigidity Conjecture) The Bloch group B(C) is countable. Given an ideal triangulation of N with cross-ratio parameters z1 0,...,z0 n, the Bloch invariant of N, β(n) = [zi 0 ]. In [30], Neumann and Yang proved that β(n) B(C) and is independent of the choice of ideal triangulation of N. The Bloch regulator map, ρ : B(C) C/π 2 Q relates β(n) to the volume and Chern-Simons invariant of N. For (z,t) P 0 (N), let N(z) denote the manifold obtained by gluing n ideal tetrahedra with parameters z 1,...,z n and the same gluing pattern as N. Then the Bloch invariant of N(z) is still defined but belongs to P(C) in general and belongs to B(C) when z corresponds to a (p,q) Dehn surgery. Let β N (z) = β(n) β(n(z)). We call β N the variation of the Bloch invariant. N is a function from P 0 (N) to P(C). Variation of the volume and Chern-Simons is similarly defined. It is shown implicitly in [31] that A 0 (l,m) determines the variation of volume of N and implicitly in [42] that A 0 (l,m) determines the variation of the Chern-Simons invariant of N. A well known conjecture from K-theory states that: Conjecture (Ramakrishnan Conjecture) The Bloch regulator map ρ is injective. In view of Ramakrishnan s Conjecture it is natural to make the following conjecture: Conjecture A 0 (l,m) determines the variation of the Bloch invariant. We show:

5 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 5 Theorem If two one-cusped hyperbolic 3-manifolds have the same A 0 (l,m) then the difference of the variation of their Bloch invariant has image in B(C). Since β N is defined for values in P 0 (N), in view of the Bloch Rigidity Conjecture we expect that the difference in variation is constant and hence determined by A 0 (l,m). We bypass the Bloch Rigidity Conjecture in the case when A 0 (l,m) is an equation of a rational curve. We show: Theorem A 0 (l,m) determines the variation of the Bloch invariant if A 0 (l,m) is an equation of a rational curve. Let N(p,q) denote the manifold obtained by Dehn filling a (p,q) curve on the torus boundary of N. Symmetries of A 0 (l,m) translate to symmetries of the Bloch invariant. We show: Theorem If A 0 (l,m) is an equation of a rational curve and A 0 (l,m) = A 0 (l a m b,l c m d ) then β(n(p,q)) = β(n(ap + bq,cp + dq)). Acknowledgements: This paper is part of my Ph.D. thesis at Columbia University. It is my pleasure to thank my advisor Walter Neumann for his guidance, encouragement and patience. 2. The PSL A-Polynomial Definitions in the SL(2, C) case. Let N be a 3-manifold and let G = π 1 (N). Then the SL(2, C) representation variety of N is defined as R(N) = Hom(G, SL(2, C)). Since N is a 3-manifold its fundamental group G is finitely presented. Let G = g 1,...,g n : r 1,...r m be a presentation of G. Using this presentation it is easy to see that R(N) C 4n and is a solution of polynomial equations, one for every generator, which makes the determinant of its image equal 1, and four for each relator. Hence R(N) is a complex affine algebraic set. We are interested in conjugacy classes of representations. SL(2, C) acts on R(N) by conjugation: for any A SL(2, C) and for any representation ρ R(N) we can define A ρ = i A ρ where i A is the inner automorphism X AXA 1. This action is algebraic and the character variety X(N) is defined to be the algebro-geometric quotient of this action. There is a direct way to see the character variety. For each g G, define I g : R(N) C by setting I g (ρ) = trace(ρ(g)) for every representation ρ R(N). Then I g C[R(N)], the coordinate ring of R(N). Let the trace ring T(G) be the sub-ring of C[R(N)] generated by all the functions I g for g G. The following is shown in [14] ] (see also [35]):

6 6 Abhijit Champanerkar Proposition 2.1. Suppose that the group G is generated by elements g 1,...,g n. Then the trace ring T(G) is generated by the elements I V, where V ranges over all elements of the form g i1... g ik with 1 k n and 1 i 1 <... < i k n. (Note that this set of generators of T(G) has 2 n 1 elements.) Using the above proposition we can describe the character variety explicitly. Let N = 2 n 1 and let V 1,...,V N be words of the above form in some order. Define a map t : R(N) C N by t(ρ) = (I V1 (ρ),...,i VN (ρ)). Then X(N) is parameterized by the image t(r(n)) C N and will be identified with the image t(r(n)). The map t : R(N) X(N) is algebraic and surjective and it is shown in [14] that the image t(r(n)) is an algebraic set. A more elementary proof of the fact that X(N) is an algebraic set is given in [21]. It is also shown in [21] that the trace ring T(G) defined above is generated by the smaller set {I V } where V {g i : 1 i n} {g i g j : 1 i < j n} {g i g j g k : 1 i < j < k n}. This set contains n(n 2 + 5)/6 elements. The above definitions hold for any finitely generated group. Character varieties are used to detect essential surfaces in 3-manifolds, as described in [14]. An ideal point of a curve C in X(N) gives a valuation on the function field F of C. This valuation gives an action of SL(2,F) on a tree via the Bass-Serre-Tits theory of trees. This action can be pulled back to an action of π 1 (N) on a tree. The action of π 1 (N) on a tree can be used to construct essential surfaces in N using a construction by Stallings- Waldhausen. Let N be a 3-manifold with torus boundary. Then we can study the representations of π 1 ( N) in SL(2, C) which extend to representations of π 1 (N) in SL(2, C). Let us fix a basis B = {L,M} of π 1 ( N) = Z Z. The inclusion map π 1 ( N) into π 1 (N) induces the restriction map r : X(N) X( N). Let R( N) be the subvariety consisting of diagonal representations. Let p B : C C be defined as follows: if ρ is a representation defined by ρ(l) = ( ) l 0 0 l 1 and ρ(m) = ( ) m 0 0 m 1 then p B (ρ) = (l,m). It follows that p B is an isomorphism. Also the map t : R( N) X( N) defined above restricts to a surjection t : X( N) which is generically 2-to-1. We have: X(N) r X( N) t 2:1 p B 1:1 C C

7 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 7 Let X (N) be the union of irreducible components Y of X(N) such that the closure of r(y ) is 1-dimensional. For each component Z of X (N) let Z be the curve t 1 (r(y )). Definition 2.1. The curve D N is the union of curves Z as Z varies over all the components of X (N). The defining polynomial of the closure of the image of D N in C C is called the A-polynomial of N and denoted by A N (l,m). Remark The curve D N consists of the eigenvalues of the meridian and longitude of representations of π 1 ( N) in SL(2, C) which extend to π 1 (N) and is also referred to as the eigenvalue variety of the 3-manifold N. 2 A defining polynomial vanishes exactly on the curve and has no repeated irreducible factors. Such a polynomial is unique up to multiplication by non-zero constants and powers of l and m. 3 A N (l,m) depends upon the choice of the basis B of π 1 ( N). If B 1 = {L a M b,l c M d }, where det ( a b c d) = 1, is a different basis, then A B 1 N (l,m) =. A B N (la m b,l c m b ) where. = means equality up to factors of m and l.. ( (4) A N (l,m) = A N (1/l,1/m) i.e. 1 0 ) 0 1 is always a symmetry of A N (l,m) Definitions in the PSL(2, C) case. Let us define the PSL(2, C) counterparts of the above. We will denote the PSL(2, C) counterparts by an over-line on their respective SL(2, C) notation. The PSL(2, C) representation variety of N is defined as R(N) = Hom(G,PSL(2, C)). Since PSL(2, C) SO(3, C) SL(3, C), one can see as before that R(N) C 9n and is a complex affine algebraic set. Similar to the SL(2, C) case, there is a PSL(2, C) action on R(N) by conjugation and the PSL(2, C) character variety X(N) is defined to be the algebro-geometric quotient of this action. There is a surjective quotient map t : R(N) X(N) which is constant on conjugacy classes of representations. Let us denote t(ρ) by χ ρ. The analogous result to Proposition 2.1 is proved in [3]. To describe it let m = n(n 2 + 5)/6 and let {y 1,...,y m } = {g i : 1 i n} {g i g j : 1 i < j n} {g i g j g k : 1 i < j < k n}. Let F n denote the free group on the symbols x 1,...,x n. Proposition 2.2. Let ρ, ρ R(N). Choose matrices A 1,...,A n, B 1,...,B n SL(2, C) satisfying ρ(g i ) = ±A i and ρ (g i ) = ±B i for each i. Define ρ,ρ R(F n ) by requiring that ρ(x i ) = A i and ρ (x i ) = B i for each i. Let y 1,...,y m be the n(n 2 + 5)/6 elements of F n associated to generators x 1,...,x n as described above. Then χ ρ = χ ρ if and only if there is a homomorphism ǫ Hom(F n, {±1}) for which trace(ρ (y j )) = ǫ(y j )trace(ρ(y j )) for each j {1,...,m}.

8 8 Abhijit Champanerkar The quotient map Φ : SL(2, C) PSL(2, C) induces an algebraic map Φ : R(N) R(N). This map is not onto in general. The condition whether or not an element in R(N) lifts is well understood. See [3] for more details and examples of N such that Φ is not onto Definition of the PSL(2, C) A-polynomial. Let N be a 3-manifold with torus boundary. Let us define the PSL(2, C) analog of the A-polynomial. The inclusion of π 1 ( N) into π 1 (N) induces the restriction map r : X(N) X( N). Let R( N) be the subvariety consisting of diagonal representations. Let p B : C C be defined as follows: if ρ is a representation defined by ( ) l 0 ρ(l) = ± 0 l 1 ( ) m 0 and ρ(m) = ± 0 m 1 then p B (ρ) = (l 2,m 2 ). It follows that p B is an isomorphism. The map t : R( N) X( N) defined above restricts to a surjection t : X( N) which is generically 2-to-1. We have: X(N) r X( N) t 2:1 p B 1:1 C C Let X (N) be the union of irreducible components Y of X(N) such that the closure of r(y ) is 1-dimensional. For each component Z of X (N) let Z be the curve t 1 (r(y )). Definition 2.2. The curve D N is the union of curves Z as Z varies over all the components of X (N). The defining polynomial of the closure of the image of D N in C C is called the PSL(2, C) A-polynomial of N and denoted by A N (l,m). Remark 2.2. A(l,m) defines the curve given by the squares of the eigenvalues of the meridian and longitude of representations in PSL(2, C) of π 1 ( N) which extend to π 1 (N). We will be interested in the case when N is a complete, orientable, finite volume one-cusped hyperbolic 3-manifold. The cusp on N is homeomorphic to T 2 [0, ) and N is homeomorphic to the interior of a 3-manifold with torus boundary. Although N is non-compact by N we will mean the torus T 2 0. The universal cover of N is the hyperbolic 3-space H 3. The covering translations are isometries of H 3 and hence lie in the group PSL(2, C). This gives us a representation of π 1 (N) into PSL(2, C) which is faithful and discrete. It is often referred to as the representation associated to the hyperbolic structure of N and is denoted by ρ 0. It follows

9 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 9 from Mostow Rigidity that any other discrete faithful representation is conjugate to ρ 0. It is a theorem of Thurston that ρ 0 lifts to a representation ρ 0 into SL(2, C). We will denote the irreducible component of X(N) containing the character of ρ 0 by X 0 (N). Similarly we will denote the irreducible component of R(N) containing ρ 0 by R 0 (N) and the image t(r 0 (N)) in X(N) by X 0 (N). The corresponding curves and factors of the SL(2, C) and PSL(2, C) A-polynomials are also denoted by the subscript 0. Let us make an observation about the relation between the SL(2, C) and PSL(2, C) A 0 -polynomials. The projection π : SL(2, C) PSL(2, C) induces the map π : X 0 (N) X 0 (N). Culler [12] showed that this map is surjective. Let h : C C C C be defined by h(x,y) = (x 2,y 2 ). Then X 0 (N) π r X( N) π p 1 B t 2:1 C C h X 0 (N) r X( N) p B 1 t 2:1 C C Let D 0 and D 0 denote the curves defined by A 0 (l,m) and A 0 (l,m) respectively. From the above diagram it follows that h(d 0 ) = D 0. So we get h 1 (D 0 ) = D +,+ 0 D +, 0 D,+ 0 D, 0 where D ±,± 0 = curve given by A 0 (±l, ±m). Hence A 0 (l 2,m 2 ) A 0 (l,m)a 0 (l, m)a 0 ( l,m)a 0 ( l, m) It is shown in [7] that a homomorphism ρ : π 1 (N) Z 2 = {±1} C which restricts non trivially to π 1 ( N) induces an involution on the SL(2, C) A- polynomial. For example in knot complements the Hurewicz homomorphism composed with the quotient map from Z to Z 2 maps the standard meridian to 1 and standard longitude to 1 giving A 0 (l,m) = A 0 (l, m). In this case A 0 (l 2,m 2 ) = A 0 (l,m)a 0 ( l,m). This relation or a similar one (e.g. A 0 (l 2,m 2 ) = A 0 (l,m)a 0 ( l, m) for m11) holds for many manifolds in the SnapPea s [40] census of cusped hyperbolic 3-manifolds. For the manifold m208 we get all the 4 factors. The curve X 0 (N) is also studied in [24]. 3. Combinatorics of ideal triangulations Ideal Tetrahedron. An ideal tetrahedron is a geodesic hyperbolic tetrahedron with all its vertices on the sphere at infinity of the hyperbolic 3-space. Let H 3 = {(x,y,z) R 3 : z > 0} denote the upper-half space model of the hyperbolic 3-space. The group of orientation preserving isometries of H 3 is identified with the group PSL(2, C). The sphere (or boundary) at infinity of

10 10 Abhijit Champanerkar H 3 denoted by H 3 is the one point compactification of the x-y plane and is identified with C = CP 1. A horosphere centered at a point p H 3 is a hypersurface in H 3 such that all geodesics with one endpoint at p are orthogonal to it. The horosphere inherits an Euclidean metric from H 3. For instance, the horospheres at infinity in the upper half space model are the planes {(x,y,z) : z = constant > 0}. The Euclidean metrics on the horosphere obtained by different values of z are constant multiples of each other. See [2] or [33] for more details. Given an oriented ideal tetrahedron T in H 3, the horosphere centered at any of its vertices cuts out an Euclidean triangle which is well defined up to similarity. Similarity classes of Euclidean triangles are parameterized by the complex upper-half plane by arranging any Euclidean triangle to have vertices 0, 1 and z, where z is in the upper-half plane, using Euclidean similarities (i.e. rotations, translations and dilations). The numbers z, 1 (1/z) and 1/(1 z) give the same triangle depending on which vertices go to 0 and 1. Moreover using the Z 2 Z 2 symmetry of the ideal tetrahedron we can see that opposite edges have the same dihedral angles and hence the Euclidean triangles cut by horospheres centered at each vertex are similar. Hence an oriented ideal tetrahedron is described completely up to (oriented) hyperbolic isometry by a single complex number z in the upper half plane. The complex number 1/z describes the same tetrahedron with opposite orientation. To specify z uniquely we must pick an edge of T, the dihedral angle at this edge will be arg(z). To each edge of T is associated one of the numbers z, 1 (1/z) and 1/(1 z), called the modulus of the edge, with opposite edges having the same modulus (see Figure 1). Once we fix an edge of the tetrahedron we will write T = T(z). Observe that the modulus of the edges are of the form ±z ǫ 1 (1 z) ǫ 2 where ǫ i { 1,0,1}. In the upper-half space model of H 3, the vertices of an ideal tetrahedron are extended complex numbers, say z 1, z 2, z 3, z 4. The modulus of the edge with ideal vertices z 1 and z 2 can also be obtained as the cross-ratio [z 1 : z 2 : z 3 : z 4 ] of the vertices, where the cross-ratio is defined as: [z 1 : z 2 : z 3 : z 4 ] = (z 3 z 2 )(z 4 z 1 ) (z 3 z 1 )(z 4 z 2 ) Hence the edge moduli are also referred to as cross-ratio parameters. An ideal tetrahedron is said to be flat if the cross-ratio has imaginary part 0 and degenerate if two or more of its vertices are identified. In the later case the cross-ratio parameters equal 0, 1 or. In all other cases the ideal tetrahedra is said to be non-degenerate. The volume of an ideal tetrahedron is a function of its cross-ratio parameter. vol(t(z)) = D 2 (z)

11 A-polynomial and Bloch invariants of hyperbolic 3-manifolds z z 1 1 z z 1 (1 z) 1 (1 z) 1 (1 z) z z z (a) (b) Figure 1. (a) Euclidean triangle cut out by horosphere. (b) Ideal tetrahedron with edge moduli where D 2 (z) is the Bloch-Wigner dilogarithm defined as: D 2 (z) = Imln 2 (z) + log z arg(z), z C {0,1} where ln 2 (z) is the classical dilogarithm function defined as: z n ln 2 (z) = n 2 ( z 1) (3.1) n= Ideal Triangulations. An ideal triangulation of a manifold N is a cell complex X formed by gluing tetrahedra along faces in which the link of every vertex is a torus and N is homeomorphic to X X (0). The vertices of the ideal triangulation can be visualized to be at infinity with torus cusps. Let N be a cusped hyperbolic 3-manifold with an ideal triangulation T with n ideal tetrahedra 1,..., n. N is homeomorphic to the interior of a compact 3-manifold with torus boundary. Since N has Euler characteristic zero the number of edges of N is equal to the number of tetrahedra. We number the tetrahedra by an index i and the edges by an index j where both i and j run from 1 to n. Assume that each i is an hyperbolic ideal tetrahedron and let i = (z i ). At every edge of N the tetrahedra abutting the edge close up as one goes around the edge. Since the edge moduli at any edge look like ±z ǫ 1 (1 z) ǫ 2 for ǫ i { 1,0,1}, the edge moduli of these tetrahedra satisfy a gluing condition of the form n z r ji(1 z) r ji = ±1 (j = 1,2,... n) (3.2) These are called the gluing equations. The exponents r ij and r ij can be integers other than ±1 as more than one edge of a single tetrahedron can be

12 12 Abhijit Champanerkar identified. Any solution to the above equations ensures that the hyperbolic metric is well defined around an edge and hence gives a hyperbolic metric on N which is in general incomplete. Each cusp of N is homeomorphic to T 2 [0, ). The cusp torus inherits a triangulation by triangles cut out by the horoshperes centered at vertices of the ideal tetrahedra. Since the Euclidean metric on the horospheres is defined only up to scale, these triangles are well defined only up to Euclidean similarities. Hence the cusp torus gets a similarity structure from its triangulation. This similarity structure can be realized as a holonomy representation of π 1 ( N) into the group of Euclidean similarities which is isomorphic to Aff(C) = {az + b : a 0, b C}. If we fix a basis {L k, M k } of the k-th cusp torus then the derivative of the image of the basis in the holonomy representation, which we denote by l k and m k, can be written in terms of the cross-ratio parameters in the following way: l k = ± n ki(1 zl z) l ki m k = ± n ki(1 zm z) m ki k = 1,... h (3.3) For N to be complete the similarity structure on every cusp should be Euclidean. This means that m k = l k = 1 at every cusp. This condition gives two more equations at every cusp called the completeness equations. A solution z 0 = (z1 0,...,z0 n ) to Equations 3.2 to 3.3 with the condition that each zi 0 is in the upper-half plane or each zi 0 is in the lower half plane, gives a complete, finite volume hyperbolic structure on N. We will refer to the parameter for the complete hyperbolic structure on N to mean the parameter z 0 with each z i in the upper-half plane. We call an ideal triangulation geometric if the gluing and completeness equations have a solution with each z i in the upper half plane. A geometric ideal triangulation gives a hyperbolic structure on N. In this paper by an ideal triangulation we mean a geometric ideal triangulation. It follows from Mostow rigidity that the set of solutions to the above equations which give the parameters for the complete structure is a discrete set. Let r r 1n r r 1n R =.... (3.4) r n1... r nn r n1... r nn be the n 2n matrix consisting of the powers r ji, r ji. This encodes the gluing conditions. Similarly we have a h 2n matrix L consisting of the powers l ki, l ki and h 2n matrix M consisting of the powers m ki, m ki which encode the completeness equations. The matrix U = L M R (n+2h,2n) (3.5)

13 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 13 is called the gluing matrix for the manifold N. See [31] for more details on the combinatorics of triangulation and properties of the gluing matrix U Parameter space and Holonomies. Let N be a hyperbolic 3-manifold with an ideal triangulation T. Let z 0 = (z 0 i,...,z0 n) be the parameter for the complete hyperbolic structure on N. Definition 3.1. Let P T (N) = {(z,t) C n C : z satisfies Equation 3.2 and tπ n z i (1 z i ) = 1} We call P T (N) the parameter space of N with respect to the triangulation T. Let P T 0 (N) denote the component of P(N) which contains z0. Remark 3.1. (1) P T (N) can be seen as the space of all hyperbolic structures on N induced by the triangulation T. (2) We drop the superscript T when the context is clear. We will show in Section 3 that P0 T (N) is independent of the ideal triangulation. (3) The coordinate t ensures that the parameters do not degenerate to 0 or 1. We will drop the coordinate t in later discussions. z P(N) will mean (z,t) with t defined with the above equation. For z P(N), let N(z) denote the manifold obtained from gluing n tetrahedra with parameters z 1,...,z n and with the same gluing pattern as N. N has a hyperbolic metric which is in general incomplete and is complete when z = z 0. The volume of N is well defined and is the sum of the signed volumes of the individual tetrahedra, the sign being positive if z i is in the upper-half space and negative if z i is in the lower-half space. It is shown in [31] and [38] that P 0 (N) is smooth at the point z 0 and has dimension h. Let Def(N) be a small neighborhood of z 0 contained in P 0 (N) and call it the deformation space. Also let Φ : Def(N) C h, be defined by Φ(z) = (log(m 1 (z)),...,log(m h (z))) Φ maps Def(N) P 0 (N) biholomorphically to a neighborhood of 0 C h. Let D be a neighborhood of 0 C h onto which Def(N) is mapped by Φ. For z Def(N), let N(z) denote the completion of N(z) with the hyperbolic metric given by the parameter value z. N(z) differs from N topologically by the addition of a set γ k of limit points at the k-th cusp. γ k can be empty, a point or a circle. When γ k is empty the cusp is left unsurgered, when γ k is a point N(z) is not a manifold and when γ k is a circle N(z) is homeomorphic to a manifold obtained by doing a topological Dehn surgery on the k-th cusp. In the last case either the hyperbolic metric is singular along γ k or N(z) is a complete hyperbolic 3-manifold. In case N(z) is a complete hyperbolic 3-manifold then there exist co-prime integers p k and q k such that p k log(m k (z)) + q k log(l k (z)) = 2πi and N(z) is

14 14 Abhijit Champanerkar obtained by (p k,q k )-Dehn surgery on the k-th cusp of N. We let (p k,q k ) = if the cusp is left unsurgered. m(z) = l(z) = 1 if the cusp is left unsurgered and m(z) p l(z) q = 1 if the cusp is surgered along a (p,q) curve on the k-th cusp torus on N. Let us define a degree one ideal triangulation for a hyperbolic 3-manifold. Gluing finitely many tetrahedra by identifying all the 2-faces in pairs gives a cellular complex X which is a manifold except possibly at the vertices. If the complement of the bad points is oriented then X is called a geometric 3-cycle. In this case the complement X X (0) of the vertices is an oriented manifold. Let N be a hyperbolic 3-manifold. A degree one ideal triangulation of N consists of a geometric 3-cycle X and a map f : X X (0) N such that: (1) f is degree one almost everywhere in N. (2) For each tetrahedron S of X there is a map f S to an ideal tetrahedron in H 3 CP 1, bijective on vertices, such that f S S (0) : S S (0) N is the composition π f S S S (0), where π is the projection π : H 3 N. Degree one ideal triangulations are much more general than what we will need. If M is a hyperbolic 3-manifold obtained from Dehn surgery on some of the cusps of a cusped manifold N then the ideal triangulation of N deforms to a degree one ideal triangulation of M. Let N be a one-cusped hyperbolic 3-manifold ideally triangulated with n tetrahedra. Let l = l 1 and m = m 1. Define Hol : P(N) C C, by Hol(z) = (l(z),m(z)) Hol (for holonomy) is a rational function on P(N). Let Z = Y i where Y i is a component of P(N) whose closure of the image under Hol is a curve in C C. We define Definition 3.2. The image Hol(Z) is called the Holonomy variety with respect to the triangulation T and denoted by H T (N). The defining polynomial of the closure of H T (N) is denoted by H T (l,m). Let H0 T (N) (respectively H0 T (l,m)) denote the image Hol(P 0 T (N)) (respectively factor of HT (l,m)). Remark 3.2. (1) We drop the superscript T when the context is clear. We will show in Section 3 that H0 T (N) is independent of the ideal triangulation. (2) H(l,m) =. H(1/l,1/m) i.e. ( ) is always a symmetry of H(l,m). Let us describe a generalization of H(l,m) to more than one cusps. Let N be a hyperbolic 3-manifold with h > 1 cusps. We can define the holonomy map Hol : P(N) C 2h by Hol(z) = (l 1 (z),m 1 (z),...,l h (z),m h (z)). We know that P 0 (N) has dimension h. Let Q k P 0 (N) be the subset consisting of

15 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 15 parameters corresponding to leaving all but the k-th cusp unsurgered. Q k is obtained by adding h 1 completeness equations, one for each unsurgered cusp, to the gluing equations. Hence Q k is a curve in P 0 (N). Let p k : C 2h C 2 be the map p k (z 1,w 1,...,z h,w h ) = (z k,w k ). Then p k Hol(Q k ) is a curve in C 2. We call this curve the k-th Holonomy variety of N and denote it by H0 k(n). The defining polynomial of the closure of Hk 0 (N) is denoted by H0 k(l,m). 4. The developing map In this section we define a map D : P(N) X(N). Using this map we show that the polynomial H(l,m) defined in Section 2 divides the PSL(2, C) A- polynomial. Furthermore we show that the the curves P 0 (N), H 0 (N) and the polynomial H 0 (l,m) is independent of the ideal triangulation Construction of the developing map D. Let N be a hyperbolic 3- manifold with an ideal triangulation T with n tetrahedra σ 1,...,σ n. Let z 0 = (z1 0,...,z0 n ) be the parameter for the complete hyperbolic structure on N. Fix a base point x N such that x interior (σ 1 ). Let Ñ be the universal cover of N and let p : Ñ N be the covering map. The triangulation T lifts to a triangulation T on Ñ. Fix a base point x Ñ such that p( x) = x and fix a lift σ 1 of σ 1 which contains x. The triangulation T is π 1 (N)-equivariant i.e. it is invariant under the π 1 (N) action on Ñ and if ỹ = g x for some g π 1 (N) then ỹ interior g( σ 1 ). For any parameter z = (z 1,...,z n ) P(N) we construct a map φ : Ñ H 3 inductively as follows: Send σ 1 to the ideal tetrahedra with vertices z 1, 1, and 0. Observe that [z 1 : 1 : : 0] = z 1 and hence the cross-ratio parameter of φ( σ 1 ) is z 1. Assume φ is defined for a triangulated subset S of Ñ. Let σ be a lift of σ i for some i for which σ has a face in common with S. Let a, b, c, d be the vertices of σ and σ S equal the face of σ with vertices a, b and c. Define φ( σ) to be the unique hyperbolic tetrahedra with vertices φ S (a),φ S (b),φ S (c) and w such that the cross-ratio [φ S (a) : φ S (b) : φ S (c) : w] = z i. Observe that φ( σ) is well defined even if all the four vertices are in S as the z i s satisfy the gluing equations. The map φ is well-defined and continuous and is called the developing map for the hyperbolic structure (in general incomplete) induced on N by the parameter z = (z 1,...,z n ). Let us see how the map φ changes if we change the image of σ 1. Let φ be the developing map defined using another choice of the image of σ 1. Let φ ( σ 1 ) = and let a, b, c and d be the vertices of such that [a : b : c : d] = z 1. Since and have the same cross-ratio there is a unique hyperbolic isometry α such that α( ) =. Hence φ ( σ 1 ) = α(φ( σ 1 )). Since

16 16 Abhijit Champanerkar φ and φ are defined inductively we get: φ ( σ) = α(φ( σ)) (4.1) for any σ T and hence φ (x) = α(φ(x)) for all x Ñ. The map φ gives rise to the representation ρ : π 1 (N) PSL(2, C) as follows: For any g π 1 (N) let σ g = g σ 1. By definition of φ, the cross-ratio parameters of φ( σ 1 ) and φ( σ g ) are the same and hence there is a unique orientation preserving hyperbolic isometry τ which takes φ( σ 1 ) to φ( σ g ). Define ρ(g) = τ. The relation between φ and ρ is: φ(g σ 1 ) = ρ(g)(φ( σ 1 )) (4.2) Using 4.1 it is clear that this gives a representation of π 1 (N) into PSL(2, C). If ρ is the representation obtained from φ by changing the image of σ 1 then using 4.1 and 4.2 we can see that: ρ(g) = α 1 ρ (g)α for all g π 1 (N) (4.3) Let φ be the developing map defined using a different base point, say x σ i, where σ i is a lift of σ i and φ ( σ i ) is the ideal tetrahedron with vertices z i, 1, and 0. Since φ( σ 1 ) and φ ( σ i ) have 3-vertices in common, the induced representations ρ and ρ are the same. Hence a parameter z P(N) gives a conjugacy class of representations in PSL(2, C) and hence a well defined element of X(N). This representation is called the holonomy representation associated to the hyperbolic structure on N induced the parameter z. Let φ z be the developing map obtained as above starting with the image of σ 1 to be the tetrahedra with vertices z 1, 1, and 0 and let ρ z denote the induced holonomy representation. Let D : P(N) X(N) be defined by D(z) = χ ρz Properties of D. D is Algebraic: We will show that D is an algebraic map. Let D : P(N) R(N) be the map defined by D(z) = ρ z. Since D = D t and t is algebraic it is enough to show that the D is algebraic. The fundamental group π 1 (N) of N is generated by face pairings of some fundamental domain R Ñ which is triangulated in T. Let σ 1,..., σ n be the tetrahedra which make up R. Any face pairing of R is a composition of a face pairings of σ i and their inverses. Note that the face pairing of each σ i may or may not belong to π 1 (N). The image ρ z (π 1 (N)) is generated by the face pairings of the tetrahedra i with vertices 0, 1, and z i for 1 i n and their inverses. All the face pairings of i are generated by the following

17 elements of PSL(2, C): A-polynomial and Bloch invariants of hyperbolic 3-manifolds 17 ( ) ( zi 0 ± 0 1/ zi 1 1/ ) z, ± i 1 z i 0 1/, ± z i 1 ± ( ) ( ) 0 1 i i, ± i zi 1 z i zi 0 1 zi 1 z i z i, (4.4) Hence the image of the representation ρ z is in the subgroup of PSL(2, C) generated by products of the above matrices and their inverses. Since PSL(2, C) acts on C 3 via the action on 2 2 traceless matrices by conjugation, we get a faithful representation Θ : PSL(2, C) SL(3, C) defined as: Θ(± ( ) a b ) = c d ad + bc ac bd 2ab a 2 b 2 (4.5) 2cd c 2 d 2 It is easy to see that det(θ(±a)) = 1 for all ±A PSL(2, C) and the image of matrices in 4.4 are matrices with entries as rational functions of z i s. Hence the map D is algebraic and hence D is algebraic. D is 2:1: In general D is not onto. We show that D is generically 2 : 1 onto its image. Lifting representations to give geometric triangulations has been studied in [7] (Section 4.5) and also in [30] (Section 8). Consider the triangulated fundamental domain R Ñ considered above and let v 1,... v s denote its vertices. Since N has one cusp, π 1 (N) acts transitively on the vertices of the triangulation T of Ñ. Moreover since the triangulation T of N is an ideal triangulation, the only vertex in N is at infinity and hence π 1 ( N) has a fixes a vertex at infinity in Ñ. Let v 1 be a vertex of σ 1 which always maps to under φ z for all z P(N). Let g 1,...,g s π 1 (N) be such that g j (v 1 ) = v j for all 1 j s. To each tetrahedron σ i of R we can associate 4 group elements such that g ij (v 1 ) = v ij for j = 1,... 4 where v i1,...,v i4 are the vertices of σ i. For any z P(N) the cross ratio parameter of φ z ( σ i ) is z i = [ρ z (g i1 )( ) : ρ z (g i2 )( ) : ρ(g i3 )( ) : ρ(g i4 )( )] (4.6) Let χ X(N) be a character of a representation ρ R(N). If the image ρ(π 1 ( N)) Z 2 Z 2 then ρ(π 1 ( N)) has a fixed point on the sphere at infinity i.e. as a subgroup of PSL(2, C), ρ(π 1 ( N)) has at least one common eigenvector. When ρ = ρ 0, the representation associated to the hyperbolic structure on N, ρ(π 1 ( N)) consists of parabolic isometries and has a single fixed point on the sphere at infinity. In general ρ(π 1 ( N)) consists of hyperbolic or elliptic isometries and has two fixed points on the sphere at infinity. We can always conjugate a representation so that one of the fixed points of ρ(π 1 ( N)) is the point. Starting with a character χ X(N) let ρ be a

18 18 Abhijit Champanerkar representation such that χ ρ = χ and ρ(π 1 ( N)) fixes. The fundamental domain can be reconstructed using the vertices ρ(g 1 )( ),... ρ(g s )( ). Similarly the cross ratio parameters of each σ i are obtained from Equation 4.6 by substituting ρ in place of ρ z. In case two or more of the vertices coincide the parameters degenerate, i.e. become 0, 1 or, the pull back is not defined. Since the g i s and π 1 ( N) generate π 1 (N), the image under D of the pull back of χ goes to itself. Since generically ρ(π 1 ( N)) has two fixed points the map D is generically 2 : 1. Observe that the pull back is defined for every element in the image of D. Since the condition that two or more vertices coincide gives an additional equation, the characters on the component X 0 (N) for which the pull back is not defined is 0-dimensional. Hence the image D(P 0 (N)) consists of almost all the characters on X 0 (N). Holonomies of cusp torus: Let T be the cusp torus of N. Then T inherits a triangulation by similarity classes of Euclidean triangles cut off by the horospheres centered at the vertices of the ideal tetrahedra. This gives a similarity structure on T which is Euclidean if the ideal triangulation gives the complete hyperbolic structure on N. The similarity structure gives a holonomy representation ξ : π 1 (T) Sim(E 2 ) = Aff(C) = {az + b : a,b C, a 0}. Euclidean similarities can be identified with a subset of PSL(2, C) via the identification: az + b ± ( ) a b/ a 0 1/ a Fix a basis B = {L,M} of π 1 (T) π 1 (N). Let ξ(l) = lz + b 1 and ξ(m) = mz + b 2. Then the holonomy representations of L and M are: ρ(l) = ± ( l b1 / ) l 0 1/ l and ρ(m) = ± ( m b2 / ) m 0 1/ m Hence the square of the eigenvalues of the meridian and longitude of the holonomy representation ρ z induced by the developing maps to the derivatives of the representation of the meridian and longitude in Sim(E 2 ) = Aff(C). It is proved in [38] (see also [31] Lemma 2.1) that this derivative is the number l and m from Equations 3.3. Hence we have shown: Theorem 4.1. The map D : P(N) X(N) defined by D(z) = χ ρz is algebraic, generically 2 : 1 onto its image and the following diagram commutes:

19 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 19 X(N) r X( N) D 2:1 p 1 B t 2:1 P(N) Hol C C Hence H(l, m) divides the PSL(2, C) A-polynomial A(l, m). The above diagram restricted to P 0 (N) gives that the curve H 0 (N) = p B (D 0 ). Hence we have: Theorem 4.2. H 0 (l,m) = A(l,m) and hence the curve H 0 (N) and its polynomial are independent of the ideal triangulation of N. In [16], Dunfield has shown that the map r : X 0 (N) X( N) is a birational isomorphism onto its image. Hence from the above diagram we have Theorem 4.3. Hol : P 0 (N) C C is a birational isomorphism onto its image and hence the curve P 0 (N) is birational to the curve H 0 (N) and is independent of the triangulation of N. 5. Boundary slopes One of the striking properties of the A-polynomial is that the slopes of the sides of the Newton polygon of the A-polynomial are boundary slopes of essential surfaces in the manifold. We give a similar relationship for the PSL(2, C) A-polynomial along the same lines using the PSL(2, C) character variety theory developed in [3]. Most of the proofs in Section 3 of [7] go through with minor modifications Ideal points of D N. Let N be a one-cusped hyperbolic 3-manifold. Fix a basis B = {L,M} of π 1 ( N) = H 1 ( N; Z). If S is an incompressible surface with non-empty boundary in N then S is a family of homologous simple closed curves on N and hence determines a homology class in H 1 ( N; Z) given by pm + ql. The boundary slope of S is the extended rational number p/q. Fix a curve C X(N). Then it is shown in [3] ( Section 4) that there exists a finite extension F of the function field C(C) of C and a tautological representation. P : π 1 (N) PSL(2,F)

20 20 Abhijit Champanerkar Proposition 5.1. Let Y be an irreducible component of D N. There exists a finite extension F of the function field C(Y ) of Y and a representation P : π 1 N PSL(2,F) such that ( ) ( ) l 0 m 0 P(L) = ± 0 l 1 and P(M) = ± 0 m 1 where l and m are regarded as elements of C[Y ]. Proof: By the construction of the curve D N there is a curve Y N X( N) and an irreducible component Y X(N) such that t (Y ) = r(y ) = Y N. By Section 4 of [3] there exists a finite extension F 1 of C(Y ) and a tautological representation P 1 : π 1 (N) PSL(2,F 1 ). Since the maps r and t are surjective, the function fields C(Y ) and C(Y ) are finite extensions of the function field C(Y N ). Hence we can find a common finite extension, call it F, of F 1 and C(Y ). We may regard P 1 as a representation of π 1 (N) in PSL(2,F). Since F contains C(Y ), it contains l and m. Since l and m are eigenvalues of commuting matrices P 1 (L) and P 1 (M), the representation P 1 is conjugate in GL(2,F) to a representation P satisfying the conclusion of the proposition. Proposition 5.2. To each ideal point x of D N there corresponds an incompressible surface with non-empty boundary in N. If ν is the valuation on C(D N )associated to x then the boundary slope of this incompressible surface is ν(l)/ν(m). Proof: Let Y be an irreducible component of D N and ν be the discrete valuation induced on C(Y ) by x. Let F be the finite extention of C(Y ) and P be the representation obtained using Proposition 5.1. Since F is a finite extention of C(Y ), the discrete valuation ν extends to a discrete valuation ν of F with the property that ν (f) = Nν(f) for some N and for all f C(Y ). As shown in [14] and [3] we obtain an action of π 1 (N) on the tree of PSL(2,F) determined by the representation P. Since l and m are the coordinates on D N and x is an ideal point, ν(l) or ν(m) is non-zero. This means that this action is non-trivial and Theorem 4.3 of [3] implies that N has a splitting along an essential surface with nonempty boundary. Morever Proposition 4.7 of [3] implies that the boundary of this essential surface is the unique boundary slope r such that ν (r) = 0 i.e. if r = p/q then the element γ = l q m p π 1 ( N) π 1 (N) satisfies ν (γ) = 0. Since ν is a discrete valuation ν (γ) = ν (l q m p ) = qν (l) + pν (m) = 0 One solution to this is p = ν (l) and q = ν (m) and hence the slope r = ν (l)/ν (m) = ν(l)/ν(m), since ν (f) = Nν(f) for f C(Y ). Proposition 4.7 of [3] says that this is the unique slope with this property. This

21 A-polynomial and Bloch invariants of hyperbolic 3-manifolds 21 proves our proposition Newton polygon and Puiseaux parametrizations. The Newton polygon of a polynomial F(x,y) = a mn x m y n is the convex hull of the points (m,n) Z Z where a mn 0 and is denoted by N F. To relate the boundary slopes of essential surfaces to slopes of sides of the Newton Polygon of the A(l,m) we find valuations which arise from the slopes of the sides of Newton polygons. This is done using Puiseaux parametrization. Let F(x,y) = m,n a mnx m y n = 0 be the defining polynomial of an irreducible plane curve C C C. Let S be a side of N F having slope p/q with p, q > 0 and lying below N F. Let S lie on the line with equation px+qy = d. Assume that F has no factors of x or y. Consider the following part of F: G 1 (x,y) = a mn x m y n (5.1) pm+qn=d Substituting x = x p 1 and y = txq 1 in G 1 we get: a mn x pm 1 tn x nq 1 = a mn x pm+nq 1 t n = x d 1 pm+qn=d pm+qn=d pm+qn=d a mn t n If F(x,y) = 0 then G 1 (x,y) = 0 and this happens when t is substituted by a root a 0 0 of the polynomial pm+qn=d a mnt n. Hence y 0 = a 0 x q 1 can be seen as a first approximation to solving y in terms of x for the implicit equation F = 0. One can iterate this process by substituting x = x p 1 and y = x q 1 (a 0 + y 1 ) in F to get F(x p 1,xq 1 (a 0 + y 1 )) = m,n a mn x pm+qn 1 (a 0 + y 1 ) n = x d 1F 1 (x 1,y 1 ) since S is below N F and pm + qn = d is lowest among all m and n with a mn 0. Now we can repeat this recursively with F substituted by F 1. It is a classical result that this gives a power series expansion of y which converges (see [4] or [22]). Hence we get a parametrization of the curve C of the form: x(t) = t p and y(t) = t q b n t n with b 0 0. Such a parametrization is called a Puiseaux parametrization. A Puiseaux parametrization gives us a valuation on the function field C(C) of the curve C as follows. Given a rational function k(x,y) which is non-zero in C(C), one defines the order of k to be the integer n such that k((x(t),y(t)) = t n z(t) where z(t) is a power series with a non-zero constant term. It is shown in [22] that the order is well defined and the induced function is a discrete valuation on C(C). If this valuation is ν then ν(x) = p and ν(y) = q where x and y are seen as elements of C(C). n=0

22 22 Abhijit Champanerkar In order to handle the case of the sides of N F which have non-negative slope or a side which lies above N F we change coordinates appropriately. See the proof of Proposition 3.3 of [7] for more details. From the above discussion we have Proposition 5.3. Let C be a plane curve with defining polynomial F(x, y). Assume F is not divisible by x or y. If the Newton polygon of F has a side of slope p/q then there is a valuation v on the function field of some irreducible component of C such that p/q = ν(x)/ν(y). Combining Propositions 5.2 and 5.3 we have Theorem 5.1. The slopes of the sides of the Newton polygon of A(l,m) are boundary slopes of incompressible surfaces in N which correspond to the ideal points of D N. Using Theorem 4.1 we immediately get: Theorem 5.2. The slopes of the sides of the Newton polygon of H(l,m) are boundary slopes of incompressible surfaces in N which correspond to the ideal points of H(N). 6. Computations and examples In this section we will discuss the computational aspect of H(l, m) and give examples Computation of H(l,m). The computation of H(l,m) is reduced to classical elimination theory once the gluing and completeness equations are set up. Suppose we are given an ideal triangulation of a cusped hyperbolic 3-manifold N. Then we can compute the gluing and completeness equations using the triangulation data. Jeff Weeks program SnapPea [40] reads the triangulation data of a manifold, computes the gluing and completeness equations and computes the hyperbolic structure simply by solving these equations numerically. Using the numerical solution SnapPea computes hyperbolic invariants like volume, Chern-Simons invariant, length of shortest geodesic and topological invariants like the fundamental group, first homology group, cyclic covers. Although experimental SnapPea is very accurate and experimentally reliable. SnapPea has been used extensively to study examples of hyperbolic 3-manifolds and test conjectures. A manifold can be entered into SnapPea in many ways. A knot or a link complement can be entered by drawing a knot or link projection. SnapPea includes a census of cusped hyperbolic 3-manifolds which can be triangulated by 7 or less ideal tetrahedra [6] which is usually referred to as SnapPea s cusped census. A manifold from the census can be loaded into SnapPea. Punctured torus